Cofinality quantifiers in Abstract Elementary Classes and beyond

TL;DR

This paper demonstrates how cofinality quantifiers, introduced by Shelah as a stronger logic than first-order, can be used to form Abstract Elementary Classes through a general framework of abstract Skolemization.

Contribution

It provides a systematic method to convert classes axiomatized by cofinality quantifiers into Abstract Elementary Classes using a unified Skolemization approach.

Findings

Classes with cofinality quantifiers can be structured as AECs

A general framework for abstract Skolemization is developed

The approach applies to a broad range of examples

Abstract

The cofinality quantifiers were introduced by Shelah as an example of a compact logic stronger than first-order logic. We show that the classes of models axiomatized by these quantifiers can be turned into an Abstract Elementary Class by restricting to positive and deliberate uses. Rather than using an ad hoc proof, we give a general framework of abstract Skolemization that can prove a wide range of examples are Abstract Elementary Classes.